sandwich measurable set

Suppose that A is subset of R (real line) with the property for every ε > 0 there are measurable sets B and C s.t. B⊂A⊂C and m(C\B)<ε
Prove A is measurable
By definition A is measurable we need to prove m(E)=m(E∩A)+m(E\A) for all E
the ≤ is trivial enough to show ≥:
Since C is measurable then m(E)= m(E∩C)+m(E\C)
≥ m(E∩A)+m(E\A)-ε (Since A is subset of C)
then move the ε to LHS and since for every ε so, let ε->0 , obtain the result.
is my solution right??? i thought i should use B and m(C\B)<ε somewhere. Could some one help me to check it??
many thanks